The formation of PBHs requires the generation of large overdensities in the early Universe, which are usually taken to be of inflationary origin. When such overdensities at horizon re-entry are larger than the threshold \(\delta _{c}\equiv \delta \rho / \rho \) , they collapse into PBHs. The specific value of the critical density is well known to be very sensitive to the profile of the EOS of the cosmic fluid at the time when the perturbations enter the Hubble radius. Therefore, different models of the thermal history evolution can produce distinct features in the PBH mass function through their imprints on \(w(T)\) . Indeed, the fraction of horizon patches that collapse into PBHs is given by [1]}
\(\beta ( M )\approx {\rm erfc}\!\left[\frac{\delta _{c}\big ( w[ T( M ) ] \big )}{ \sqrt{2} \, \delta _{\rm rms}( M )}\right],\) 
